Tannery's theorem can be used to prove that the binomial limit and the infinite series characterizations of the exponential are equivalent. Note that
Define . We have that and that , so Tannery's theorem can be applied and
Referencesedit
^Loya, Paul (2018). Amazing and Aesthetic Aspects of Analysis. Springer. ISBN9781493967957.
^Ismail, Mourad E. H.; Koelink, Erik, eds. (2005). Theory and Applications of Special Functions: A Volume Dedicated to Mizan Rahman. New York: Springer. p. 448. ISBN9780387242330.
^ abHofbauer, Josef (2002). "A Simple Proof of and Related Identities". The American Mathematical Monthly. 109 (2): 196–200. doi:10.2307/2695334. JSTOR 2695334.
January 01, 1970
tannery, theorem, mathematical, analysis, gives, sufficient, conditions, interchanging, limit, infinite, summation, operations, named, after, jules, tannery, contents, statement, proofs, example, referencesstatement, editlet, displaystyle, infty, nbsp, suppose. In mathematical analysis Tannery s theorem gives sufficient conditions for the interchanging of the limit and infinite summation operations It is named after Jules Tannery 1 Contents 1 Statement 2 Proofs 3 Example 4 ReferencesStatement editLet S n k 0 a k n displaystyle S n sum k 0 infty a k n nbsp and suppose that lim n a k n b k displaystyle lim n to infty a k n b k nbsp If a k n M k displaystyle a k n leq M k nbsp and k 0 M k lt displaystyle sum k 0 infty M k lt infty nbsp then lim n S n k 0 b k displaystyle lim n to infty S n sum k 0 infty b k nbsp 2 3 Proofs editTannery s theorem follows directly from Lebesgue s dominated convergence theorem applied to the sequence space ℓ 1 displaystyle ell 1 nbsp An elementary proof can also be given 3 Example editTannery s theorem can be used to prove that the binomial limit and the infinite series characterizations of the exponential e x displaystyle e x nbsp are equivalent Note that lim n 1 x n n lim n k 0 n n k x k n k displaystyle lim n to infty left 1 frac x n right n lim n to infty sum k 0 n n choose k frac x k n k nbsp Define a k n n k x k n k displaystyle a k n n choose k frac x k n k nbsp We have that a k n x k k displaystyle a k n leq frac x k k nbsp and that k 0 x k k e x lt displaystyle sum k 0 infty frac x k k e x lt infty nbsp so Tannery s theorem can be applied and lim n k 0 n k x k n k k 0 lim n n k x k n k k 0 x k k e x displaystyle lim n to infty sum k 0 infty n choose k frac x k n k sum k 0 infty lim n to infty n choose k frac x k n k sum k 0 infty frac x k k e x nbsp References edit Loya Paul 2018 Amazing and Aesthetic Aspects of Analysis Springer ISBN 9781493967957 Ismail Mourad E H Koelink Erik eds 2005 Theory and Applications of Special Functions A Volume Dedicated to Mizan Rahman New York Springer p 448 ISBN 9780387242330 a b Hofbauer Josef 2002 A Simple Proof of 1 1 2 2 1 3 2 p 2 6 displaystyle 1 1 2 2 1 3 2 cdots frac pi 2 6 nbsp and Related Identities The American Mathematical Monthly 109 2 196 200 doi 10 2307 2695334 JSTOR 2695334 Retrieved from https en wikipedia org w index php title Tannery 27s theorem amp oldid 1196064303, wikipedia, wiki, book, books, library,